Solver for convex optimization with exponent in the objective function

Question

My question is related to a previous one: Dedicated solver for convex problems

To minimize a convex function of the form $~f(x_i) = \left[ C + mx_i + \frac{s}{x_i+t} \right]^p $

with various parameters $~C$, $m$, $s$, $t~$ and $~p$, $~$where $~0 \le x_i \le 1~$ and $~p \ge 1$.

(By generating a graph of the function, I ensure that the function $f(x_i)$ is convex. $~$The constraints are all linear.)

Edit (5:30am GMT January 25): Apologies for not bringing this up earlier, since I assumed that it wouldn't make a difference. The variable actually is a vector $X =$ ($x_1$, $x_2$, $\cdots$, $x_n$).. The objective function is:

Minimise $~Z(X) = \sum_{i=1}^n Z_i = \sum_{i=1}^n f(x_i)$.

If each $f(x_i)$ is convex, then their sum $Z(X)$ is also convex.

My question is, which solvers are well-suited to minimize such a convex function (subject to linear constraints)?

Of all solvers at NEOS, only MINOS seems to come close, and even this is not perfect (sometimes returns errors for instances for which manually I'm able to obtain optimal feasible solutions).

Answer 1

You are missing some details for the claimed convexity of $f(x)$. I will assume that you meant to say convex on the domain $C+mx+\frac{s}{x+t} \geq 0$ and $x+t \geq 0$. In this case the epigraph of $f(x)$ has conic normal form: $$ \begin{array}{ll} &f \geq \left(C+mx+\frac{s}{x+t}\right)^p\\\Leftrightarrow &f \geq \left(C+mx+y\right)^p,\quad y \geq \frac{s}{x+t},\\\Leftrightarrow &f \geq z^p,\quad z=C+mx+y,\quad y \geq \frac{s}{x+t},\\\Leftrightarrow &f \geq |z|^p,\quad z=C+mx+y \geq 0,\quad y (x+t) \geq s. \end{array} $$

where the last deduction adds in the missing convexity details. You can now input the representation in any Disciplined Convex Programming language using one power cone for $f \geq |z|^p$ (see, e.g, https://docs.mosek.com/modeling-cookbook/powo.html#powers), and one rotated quadratic cone for $y (x+t) \geq s$ (see, e.g., https://docs.mosek.com/modeling-cookbook/cqo.html#rotated-quadratic-cones). These two cones are so simple that their presence will hardly affect the interior-point algorithm from a complexity point of view.

As a last remark, note that minimizing $\left(C+mx+\frac{s}{x+t}\right)^p$ on $p\geq 1$ is equivalent to just minimizing $C+mx+\frac{s}{x+t}$. Hence, if $f(x)$ really is all there is to your objective function, you will get the same optimal solution by just minimizing $z$. This will allow you to get rid of the epigraph variable $f$ and the power cone that constrain it.

As to the question of which solver to use with Disciplined Convex Programming, I refer to this answer to your previous question: https://or.stackexchange.com/a/2727.

Answer 2

Use Ipopt and a modeling language. For example, JuMP: https://jump.dev/JuMP.jl/stable/tutorials/nonlinear/introduction/

julia> using JuMP, Ipopt
julia> model = Model(Ipopt.Optimizer)
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: EMPTY_OPTIMIZER
Solver name: Ipopt
julia> set_silent(model)
julia> @variable(model, 0 <= x <= 1)
x
julia> C, m, s, t, p = 1, 1, 1, 0.1, 3
(1, 1, 1, 0.1, 3)
julia> @NLobjective(model, Min, (C + m * x + s / (x + t))^p)
julia> optimize!(model)
julia> value(x)
0.8999994186940709
julia> objective_value(model)
24.389000000008522
julia> (C + m * value(x) + s / (value(x) + t))^p
24.389000000008522

Answer 3

Best way is to apply log to the function as optimal solution to log of a function is the same for the function.
$ \log f = p[\log (C(x+t)+ mx(x+t)+s)-\log(x+t)]$
Then you have to get the actual solution by the exp of $\log$ base used (e, 2 or 10).

However log functions need to be transformed into auxillary variables. You can also use the exp with $p$ but these also have to expressed as additional variables using addition constraints. Here is the link to suggested piecewise solution by Gurobi. Gurobi provides constraints as model methods like addGenConstrLog() and addGenConstrExp(). These are available at Gurobi's reference guide. Solvers won't have a problem with 2nd degree $x^2$

Answer 4

You can use Mathematica to solve the optimization problem

$$ \min\limits_{x} f(x) = (C + mx + \frac{s}{x + t})^p \\ \text{ where } C = 1,m = 1,s = 1,t = 0.1,p = 3 \\ \\ \text{s.t. }\quad 0 \leq x \leq 1 $$

Clear["Global`*"];
(*Define the parameters*)
C1 = 1; m = 1; s = 1; t = 0.1; p = 3;
(Define the objective function)
objectiveFunction[x_] := (C1 + m*x + s/(x + t))^p;
(Perform the optimization)
solution = NMinimize[{objectiveFunction[x], 0 <= x <= 1}, x]
(Extract the optimal value of x)
optimalX = x /. Last[solution]
(Calculate the objective function value at the optimal point)
objectiveValue = First[solution]
(Optionally,verify the objective function value by direct 

substitution)
verificationValue = objectiveFunction[optimalX]

{24.389, {x -> 0.9}}